Analytic continuation of Dirichlet series with almost periodic coefficients

Details Analytic continuation of Dirichlet series with almost periodic coefficients Analytic continuation of Dirichlet series with almost periodic coefficients

2008-11-09

arxiv.org/abs/0811.1362v1

Mathematics

Complex Variables

15 pages

Scientific paper

We prove that an ordinary Dirichlet series with coefficients a(n)=g(n b) has an abscissa of convergence 0 if g is an odd 1-periodic, real-analytic function and b is Diophantine. We also show that if g is odd and has bounded variation and b is of bounded Diophantine type r>1, then the abscissa of convergence is smaller or equal than 1-1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and b is Diophantine, then the ordinary Dirichlet series has an analytic continuation to the entire complex plane.

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